Open Access
Add to BibSonomy
Abstract
Freeform illumination optics design with extended light sources to realize an accurate light control is very important, but still remains a challenging issue. Here, we propose a new method to design compact and efficient freeform lenses for extended sources. We employ a localized surface control strategy to directly modify the freeform surface to redistribute the light rays emitted from the extended LED source in a desired manner. By the combination of basic radiometry calculation and backward ray tracing, we obtain the irradiance distribution on the target plane and estimate the localized freeform surface to be modified. The optimization function with a Gaussian form is adopted to modify the localized surface. The smoothness of the freeform surface is taken into account in the optimization process to guarantee the processability of the freeform optics. We demonstrate the effectiveness of the proposed method with three design examples.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Conventional LED sources with planar emitting surfaces emit light rays in the upper hemisphere and generate nonuniform irradiance distributions on the illumination targets. However, for practical applications such as road lighting, indoor and outdoor lighting, uniform irradiance distributions with specific shapes are often needed. Secondary optics are usually employed to redistribute the spatial energy of LEDs to satisfy these specific lighting requirements. Here, we focus on freeform surface optics because of its superior light controlability. The basic problem of freeform illumination optics design is to solve reflective or refractive freeform optics to transform the source light distribution into a desired manner, which is one of the most important concerns in the non-imaging optics field.
Extensive research has been done on designing freeform illumination optics with point-like source (zero-étendue source) assumption where the source size is negligible compared with the freeform optics [1–7], and great achievements are constantly being exposed. However, for the compactness and lightweight illumination systems, where the LED size is no longer ignorable compared with the optical system [8], the freeform illumination optics design becomes complicated. Besides, the widespread LEDs usually have 180$^{\circ }$ opening angles, achieving a compact illumination system that can collect almost all energy from LEDs with an accurate light control gets even more challenging. Up to now, there have been several methods devised for extended light sources. Simultaneous multiple surfaces (SMS) method allows to simultaneously construct multiple freeform surfaces to couple edge wavefronts from the corners of the LED emitter to the corresponding wavefronts of the illumination pattern. This wavefront tailoring method formulates the extended LED source design problem as coupling input and output edge wavefronts, which can design compact freeform lenses delivering flat-top illumination patterns [9–11]. Feedback design method uses an optics model obtained by the point-like source assumption as the initial solution, and then updates the target illumination with a feedback function to iteratively improve the design. The difference between the actual illumination and the desired one is employed to define the feedback function during each iteration [12–16]. The optimization-based method firstly represents the freeform surfaces using the basis functions, and then optimizes their coefficients to minimize the deviation between the real illumination and the prescribed one [17–21]. The deconvolution design method iteratively reduces the blur effect caused by the extended light sources with the deblurring method [22–24], which has the potential to achieve super compact illumination optics. Recently, Brand and Birch put forward an impressive idea for controlling beam dilations that dilute the intensity of the irradiance by locally modifying the curvature of the optical surface [25]. Specifically, they use an edge ray mapping to achive polygonal targets and improve the illumination uniformity by adding some appropriate saddle surfaces to the freeform surface [26].
In this paper, we propose another approach for controlling localized freeform surface for uniform irradiance distributions with extended LED sources. Firstly, the combination of basic calculation of radiometry and backward ray tracing is adopted to obtain irradiance generated by the freeform surface. Then, we determine the localized freeform surface that contributes irradiance of the specific checking point on the illumination target via backward ray tracing method. After that, the irradiance of this target point is changed by adjusting the curvature distribution of the corresponding localized surface. We also pay attention to the smoothness of the freeform surface for convenient manufacturing, this is realized via imposing an envelope function on the surface modifying process. We call this method as localized surface control (LSC) method. The whole design process that combines the design idea presentation, the localized surface determination and the optimization strategy is detailed in Section 2. In Section 3, we provided several challenging design examples including simulations and experimental tests to prove the feasibility of the proposed method. In Section 4, a brief conclusion is presented.
2. Design method
2.1 Formulation of the idea
The purpose of this research is to design a compact freeform lens with high energy collecting efficiency that can produce a uniform irradiance $E_{\text {0}}$ on a given target plane from an extended LED source, as shown in Fig. 1(a). The center of the extended LED source is located at the origin, the entrance surface of the lens is an analytical surface and the exit surface is a freeform one. According to the knowledge of radiometry, the irradiance of a specific point on the target can be directly determined by the corresponding incoming beam from the extended light source [27,28]. Specifically, the irradiance is calculated as $E=\int L{\text {cos}}\theta {\text {d}}\omega$, where $\theta$ represents the angle between a differential portion of light and the surface normal and $\omega$ is the corresponding solid angle, $L$ denotes the radiance of an infinitesimal tube of the incident light beam. For the Lambertian light source, LED, the radiance $L$ is constant along a ray path in a lossless optical system, and the irradiance of this specific target point is just determined by $\theta$ and $\omega$. In this research, we intend to change $\theta$ and $\omega$ via the LSC method to modify the freeform surface for compact and uniform illumination optics. The most critical problem in this paper is how can we change $\theta$ and $\omega$ so that the corresponding irradiance on the target changes accordingly.
Fig. 1. Illumination design for the extended LED source (a) uniform illumination system; (b) increased surface curvature causes the irradiance increasing locally; (c) decreased surface curvature leads to the irradiance decrease locally.
It is common sense that a convex lens has the ability to converge light, while the concave lens diverges the light beam. If we increase the curvature of the center freeform surface in Fig. 1(a), the irradiance of the target center increases too, as shown in Fig. 1(b). Conversely, the irradiance decreases when the curvature of the center freeform surface is reduced, this is depicted in Fig. 1(c). Based on this fact, we are going to use the LSC method to modify the curvature distribution of the freeform surface to realize the light control. Once the curvature distribution changes, the values of $\theta$ and $\omega$ change too. To make the optimization more efficient, it would be helpful if we can provide a base freeform surface that is close to the required one. For this purpose, we use the ray mapping method proposed in our previous works [3,6] to accomplish the initial lens model design. Then, two critical and intractable problems need to be solved: (1) find the localized surface that is passed by the light beam which contributes to the irradiance of the target point; (2) build an efficient modification strategy to optimize the initial freeform lens model for uniform illumination with extended LED source.
2.2 Localized surface determination
In our previous research, we have detailed the calculation process of irradiance distribution generated by the freeform lens with an extended LED source [23]. Specifically, for a specific target point on the illumination plane, the differential irradiance $\text {d}E$ contributed by a differential area of the freeform surface can be calculated as:
Fig. 2. (a)the basic calculation of radiometry for the extended source; (b) backward ray-tracing process for extended LED source; (c) determination of the localized surface.
As shown in Fig. 2(b), we reversely trace the rays from a specific target point to the freeform surface, the entrance surface and the extended source in sequence. Based on the principle of reversibility of light, if the rays intersect within the range of the extended source, the radiance will be equal to that of this ray emerging from the light source to the target point in a lossless optical system. If the ray falls outside the source domain, which means that the traced ray is invalid, the radiance will be regarded as zero. In this research, the freeform surface is described as a point cloud which is composed of a set of discrete points, and the entrance surface of the lens is an analytical one. For a given target point, we trace the rays to all of the independent points on the freeform surface from this target point, the normal vector of the freeform surface can be acquired as a solution of a nine-point difference matrix, so the intersecting points on the entrance surface and the source plane can be calculated analytically. We formulate the above backward ray-tracing process as a matrix manipulation, which can be calculated quickly in MATLAB. Therefore, the irradiance of the target point will be calculated as the accumulation of the individual radiance contributed by the valid traced light rays.
In the above ray tracing process, we collect the intersection points of the valid rays and freeform surface to determine the localized surface to be modified. The boundary of these discrete points might be an irregular one which is not convenient for the later optimization process. To address this issue, we choose a point set with a number of $m*n$ from the freeform surface in sequence, these points precisely enclose the above collected intersection points. The local surface domain composed of these chosen points is regarded as the localized freeform surface to be modified, and the boundary of this domain is a rectangular like one. Some points which have not been passed by the above valid rays are unavoidably involved in this local surface, and they are distributed in the corners of the domain, as shown in Fig. 2(c). To make the variation of these unnecessary but inevitable points in the modification process negligible, the modification function should be carefully defined, and this is will be discussed in Section 2.3.
2.3 Optimization strategy
If the calculated irradiance of a specific target point cannot meet the desired one, we find out the corresponding localized freeform surface which contributes to the irradiance firstly. Then, we change the curvature distribution of this localized freeform surface with the following modification process.
As detailed in our previous research, the information of the incident rays is parameterized by ($u$, $v$) as shown on the left side of Fig. 3, and we can regard the freeform surface as a parametric surface $\textbf {f}$ with a unit space ($u$, $v$). To facilitate the following surface modification, we further employ a spherical coordinate ($\boldsymbol {\rho },\boldsymbol {\alpha }, \boldsymbol {\beta }$) to describe the freeform surface $\textbf {f}(u,v)$, where $\boldsymbol {\rho }(u, v)$ represents the distance between the point on the freeform surface and the origin, $\boldsymbol {\alpha }(u, v)$ and $\boldsymbol {\beta }(u, v)$ denote the corresponding azimuth and elevation angles respectively. The curvature distribution of the localized surface will be changed by adjusting radial length $\boldsymbol {\rho }$ while remaining $\boldsymbol {\alpha }$ and $\boldsymbol {\beta }$ unchanged. We call the change process of surface’s curvature distribution for the desired illumination as optimization.
As illustrated in Fig. 3, the localized surface marked red corresponds to a sub-domain ($u^{'}$, $v^{'}$) inside ($u$, $v$) parametric space. The distance between the points of the localized surface and the coordinate origin can be represented as $\boldsymbol {\rho }^{'}(u^{'}$, $v^{'})$. We define the optimization function for the localized surface in $(u^{'}$, $v^{'})$ parametric space too, which is labeled green in Fig. 3. To make the optimization effective and the freeform lens machinable, the optimization function should satisfy two requirements: 1) the value of the optimization function is positive to guarantee that the freeform surface is above the light source from beginning to end; 2) the value of optimization function is continuous and equals 1 at the edge of the localized surface to ensure the smoothness of the freeform surface.
Considering these constraints, we define the optimization function as a Gaussian form, which is expressed as:
The localized curvature distribution is changed by adjusting the corresponding radial length $\boldsymbol {\rho }^{'}$, which is expressed as:
We use the root mean square (RMS) as the merit function to evaluate the uniformity of the irradiance distribution:
3. Design examples and discussions
3.1 Design examples
To verify the effectiveness of the proposed method, we provide three design examples to generate different lighting patterns with uniform irradiance distributions. The polymethyl methacrylate (PMMA) is used as the lens material, whose refractive index equals 1.49 at the wavelength of 550nm, and light sources are Lambertian LEDs. We employ the method proposed in our previous work [6] to accomplish the initial freeform lens model design. As a rule of thumb, the ratio of $h/d$ is adopted to represent the compactness of the optical system, where $h$ denotes the maximum height of the lens and $d$ is the diameter of the LED source. For all of these design examples, the sphere surface is chosen as the entrance surface of the lens, which is defined as $x^{2}+y^{2}+(z+13)^{2}=255(z\geq 0,$ unit:mm$)$. The freeform surface is composed of 200${\times }$200 points, and 80${\times }$80 sampling target points are selected to calculate the irradiance distribution on the target plane.
In the first design example, our goal is to cast a square illumination pattern with a size of 100mm ${\times }$ 100mm to show the detailed design process of the proposed method. The distance between the source and target plane is set to be 90mm. Firstly, we design the initial lens model with the point-like source assumption, the energy collection efficiency is about 97.35% without considering Fresnel loss. The height of the lens is 10.9mm, and lens size cannot be ignored compared with the whole lighting system. We use the extended LED source with a diameter of 3.74mm to calculate the irradiance distribution, the result is shown in the upper left of the Fig. 5(a). We can clearly see that the irradiance of the initial freeform lens with the extended light source shows a highly concentrated energy distribution in the central portion of the target region, which does not meet the requirement. Then, we use the LSC method to optimize the freeform surface for uniform illumination.
Fig. 5. Design of freeform lens for a square illumination pattern: (a) the convergence of the RMS; (b) irradiance distribution of the optimized freeform lens with extended LED source; (c) the normalized irradiance distributions; (d) the optimized freeform lens model with $h$/$d$ = 2.99.
At the beginning of the optimization, we change the curvature of the center part of the freeform surface. We modify this part surface with a large interval and equally spaced parameter $a$ expressed in Eq.(2) and calculate the corresponding irradiance distribution. The results show that RMS is lowest when $a = -0.004$. For the following center surface modifying process, the value of parameter $a$ varies by 10% for the new irradiance calculation. This continues until that there is a tendency that the value of RMS increases. As this illumination system is symmetric with respect to $x$ and $y$ axes, we choose the first quadrant system for the following optimization, and obtain the whole freeform surface by a mirror operation. We change other parts of the freeform surface which are determined by the target point with the highest or lowest irradiance. In the optimization process, there are two key points needed to be paid attention to: (1) as the blurring of the lighting pattern boundary cannot be avoided for extended source, we remove the illumination pattern’s edge whose irradiance is below 60% of the average irradiance to form the effective illumination area, and the checking points with highest or lowest irradiance are found in this effective area; (2) when one part of the freeform surface has been modified, but the RMS does not decrease anymore, the modification for the next localized surface starts. The detailed optimization process can be found in Fig. 4 and Fig. 5(a).
In Fig. 5(a), the localized surfaces of the freeform for some iterations are presented, and the corresponding irradiances are also provided. We can see that, as the iteration continues, the uniformity is improved gradually. RMS is reduced from 0.143 before the optimization to 0.119 after 8 localized surfaces being modified. The consumed time of this design is about 45 minutes on a Windows 10 desktop PC (Intel i7-10700K CPU with 32 GB RAM). Most of time is spent on the irradiance calculation. The finally obtained irradiance is shown in Fig. 5(b). The initial irradiance profile, the final irradiance profile and the prescribed irradiance profile of $x = 0$ mm are plotted in Fig. 5(c). The final designed lens with a maximum height of 11.1mm is shown in Fig. 5(d), which presents a compact structure with $h/d$ = 2.99.
In the second example, we intend to deliver an ellipse illumination pattern with a uniform irradiance distribution. The major axis and minor axis of the ellipse are 220mm and 140mm respectively, and the illumination distance is set as 70mm. The light source is chosen as an LED source with a diameter of 4.94mm. The irradiance of the initial freeform lens is given in Fig. 6(a), and the RMS in the desired lighting area is 0.157. The irradiance profiles of $x = 0$ mm and $y = 0$ mm are presented in Fig. 6(b). We can clearly see that there is a large gap between the initial illumination and the prescribed one. Similarly, we perform the optimization procedure for the first quadrant of the freeform surface. After 13 localized surfaces being optimized, the RMS decreases to 0.116 and trends to be stable.The final obtained irradiance distribution on the target plane is shown in Fig. 6(c) and (d), which indicate an excellent agreement with the desired one. The total computation time of this design is about 56 minutes, the energy collection efficiency is 97.11%. The height of the final obtained freeform lens is 12.406mm, the freeform surface profiles ans the lens model are shown in 6(e) and (f), the ratio of $h/d$ equals 2.51.
Fig. 6. Design of freeform lens for an ellipse illumination pattern: (a) irradiance distribution of the initial freeform lens with extended LED source; (b) the irradiance profiles on lines $x$ = 0 mm and $y$ = 0 mm; (c) irradiance distribution of the final obtained freeform lens with extended LED source; (d) the irradiance profiles on lines $x$ = 0 mm and $y$ = 0 mm; (e) freeform surface profiles;(f)the optimized freeform lens model with $h$/$d$ = 2.51.
The third design is aimed for generating a square illumination pattern in an off-axis configuration. The square target with a side length of 600mm is located at a plane 220mm away from the light source. We make the central point of the square pattern moving a distance of 80mm in +$x$ direction. A square Lambertian source with a diagonal length of 6.421mm is chosen. The calculated irradiance distribution of the initial lens model is shown in Fig. 7(a), which shows an obvious deviation between the actual illumination and the desired one, and the RMS in the predefined illumination area is 0.203. This optical system is symmetric with respect to $x$ axis, so we optimize the first and second quadrants of the freeform surface. After 27 localized surfaces being modified with 1.3 hours, the RMS is reduced to 0.116, the height of the final obtained freeform lens is 16.617mm, and $h/d$ equals 2.59. The corresponding irradiance distribution is shown in Fig. 7 (b). It clearly shows that the uniformity has been greatly improved compared with the initial one, which means that a precise light control for the compact freeform lens has been achieved in this off-axis illumination configuration with the use of LSC method.
Fig. 7. Simulation and experiment verification: (a) calculated irradiances of the initial and (b) the optimized freeform lens with extended LED source; (c) the fabricated lens prototype; (d) the experimental setup; (e) the recorded illumination pattern; (f) the comparison of the recorded and calculated irradiance profiles.
A prototype of this design is implemented as shown in Fig. 7(c), which is fabricated by injection molding, and the fabrication accuracy (Peak to Valley, PV) is about ${35\mu \text {m}}$. The light source is an LED source (LED XLAMP BLUE 465NM 1616 SMD ), and the size of the LED emitter is 4.54mm${\times }$4.54mm. The experimental setup is illustrated in Fig. 7(d) and the actual illumination pattern is shown in Fig. 7(e). The dotted lines in Fig. 7(f) represent the calculated irradiance profiles and the solid lines denote the recorded irradiance profiles. The blue lines and red lines are irradiance profiles on lines $x$ = 0 mm and $y$ = 0 mm respectively. The chromatic aberration, fabrication errors, alignment errors, and Fresnel reflection are the main error sources that contribute to performance degradation in the experiment. But overall, the experimental result agrees well with the predefined one, showing the effectiveness of the proposed method.
In the following, we illustrate the influence of the LED size on the optical performance. We increase and decrease the diameter of LED by 10% respectively, and the corresponding irradiance distributions are shown in Fig. 8. We can see that the boundaries of the illumination patterns are maintained well with different LEDs. But, the energy distributions trend to converge towards the target center when the area of LED increase. and conversely, the energy accumulates towards the edge as LED size decreases.
Fig. 8. Simulated irradiance distributions for the above three designs with different sizes of LEDs: (a) the first design; (b) the second design; (c) the third design.
3.2 Discussions
The freeform surface calculation for the point-like source is only required one time during the whole design process, which takes about 20s. The most time-consuming process of the above designs is the irradiance calculation procedure. When the freeform surface is composed of 200 ${\times }$ 200 points, the number of sampling points on the target is set as 80 ${\times }$ 80, and it takes 31.1s for one time of irradiance calculation. In our situation, the discrete points of the freeform surface are stored in matrices. And the backward ray-tracing process is looped for each target point. In each loop, we traced a matrix of rays from one target point to all points of the freeform surface. Therefore, our calculation time is less affected by the number of sampling points on the freeform surface. For other implementation of the algorithm, the features of computation time will be different. To verify the accuracy of the irradiance calculation, we perform the Monte Carlo ray tracing for the first design example, where 200 million rays are traced. The difference between two normalized irradiances is shown in Fig. 9. We can clearly see that the calculated result is in great agreement with the simulation one, which proves the reliability of the irradiance calculation process.
Fig. 9. The difference between the normalized calculation irradiance and the normalized simulation irradiance via Monte Carlo ray tracing.
Observing the above irradiance profiles, we can find there are some slight perturbations along with the irradiance profiles. This is mainly because that only small numbers of localized surfaces have been changed in this research. A smoother and more uniform irradiance might be achieved if more localized surfaces are modified, however, more time will be required. Except for changing the radial distance $\boldsymbol {\rho }$ to modify the surface curvature, other variables can also be considered simultaneously to increase the optimization freedoms, such as the azimuth and elevation angle. In this work, the entrance surface is selected as a spherical one with the center below the origin, there are two main reasons of this choice: avoiding the total internal reflection taking place on the exit surface; an analytical surface is conductive to the accurate calculation of the intersections of the light ray, the entrance surface and the source plane, making the ray tracing trajectory precisely. Besides, the modern algorithms, such as k-dimensional tree algorithm might be able to employed to accelerate the optimization process.
4. Conclusion
In conclusion, we have developed an LSC method for compact freeform illumination optics design. We use the basic calculation of radiometry and backward ray tracing method to find the critical part of the freeform surface which contributes to the illumination for the specific target point, and directly modify the curvature distribution of this localized surface to improve the optical performance. Benefit from the initial lens model with relatively acceptable optical performance, RMS which represents the uniformity converges well as the optimization going on, and the computation time is also acceptable. The superiorities of the proposed method are demonstrated by three challenging designs including simulation and experimental tests. Future work will focus on extending this method to optimize freeform illumination optics with any initial model, even for generating non-uniform illumination patterns.
Funding
Innovation Fund of WNLO; Key Research and Development Program of Hubei Province (2020BAB121); Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20190809100811375, JCYJ20210324115812035).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
References
1. L. Yang, Y. Liu, Z. Ding, J. Zhang, X. Tao, Z. Zheng, and R. Wu, “Design of freeform lenses for illuminating hard-to-reach areas through a light-guiding system,” Opt. Express 28(25), 38155–38168 (2020). [CrossRef]
2. A. van Roosmalen, M. J. Anthonissen, W. L. IJzerman, and J. Boonkkamp, “Design of a freeform two-reflector system to collimate and shape a point source distribution,” arXiv preprint arXiv:2104.04468 (2021).
3. S. Wei, Z. Zhu, Z. Fan, and D. Ma, “Least-squares ray mapping method for freeform illumination optics design,” Opt. Express 28(3), 3811–3822 (2020). [CrossRef]
4. Z. Feng, D. Cheng, and Y. Wang, “Transferring freeform lens design into phase retrieval through intermediate irradiance transport,” Opt. Lett. 44(22), 5501–5504 (2019). [CrossRef]
5. L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015). [CrossRef]
6. Z. Zhu, S. Wei, W. Li, Z. Fan, and D. Ma, “Freeform illumination optics for 3d targets through a virtual irradiance transport,” Opt. Express 29(10), 15382–15392 (2021). [CrossRef]
7. K. Desnijder, P. Hanselaer, and Y. Meuret, “Ray mapping method for off-axis and non-paraxial freeform illumination lens design,” Opt. Lett. 44(4), 771–774 (2019). [CrossRef]
8. I. Moreno and C.-C. Sun, “Led array: where does far-field begin?” in Eighth International Conference on Solid State Lighting, vol. 7058 (International Society for Optics and Photonics, 2008), p. 70580R.
9. P. Gimenez-Benitez, J. C. Mi nano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004). [CrossRef]
10. S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Mi nano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25(24), 29627–29641 (2017). [CrossRef]
11. S. Sorgato, J. Chaves, H. Thienpont, and F. Duerr, “Design of illumination optics with extended sources based on wavefront tailoring,” Optica 6(8), 966–971 (2019). [CrossRef]
12. Z. Li, S. Yu, L. Lin, Y. Tang, X. Ding, W. Yuan, and B. Yu, “Energy feedback freeform lenses for uniform illumination of extended light source leds,” Appl. Opt. 55(36), 10375–10381 (2016). [CrossRef]
13. L. Hongtao, C. Shichao, H. Yanjun, and L. Yi, “A fast feedback method to design easy-molding freeform optical system with uniform illuminance and high light control efficiency,” Opt. Express 21(1), 1258–1269 (2013). [CrossRef]
14. W. Situ, Y. Han, H. Li, and Y. Luo, “Combined feedback method for designing a free-form optical system with complicated illumination patterns for an extended led source,” Opt. Express 19(S5), A1022–A1030 (2011). [CrossRef]
15. Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for led source,” Opt. Express 18(9), 9055–9063 (2010). [CrossRef]
16. X. Mao, H. Li, Y. Han, and Y. Luo, “Two-step design method for highly compact three-dimensional freeform optical system for led surface light source,” Opt. Express 22(S6), A1491–A1506 (2014). [CrossRef]
17. E. V. Byzov, S. V. Kravchenko, M. A. Moiseev, E. A. Bezus, and L. L. Doskolovich, “Optimization method for designing double-surface refractive optical elements for an extended light source,” Opt. Express 28(17), 24431–24443 (2020). [CrossRef]
18. Z. Zhuang, P. Surman, and S. Thibault, “Multi-element direct design using a freeform surface for a compact illumination system,” Appl. Opt. 56(32), 9090–9097 (2017). [CrossRef]
19. C. Gannon and R. Liang, “Using spherical harmonics to describe large-angle freeform lenses,” Appl. Opt. 57(28), 8143–8147 (2018). [CrossRef]
20. F. Fournier and J. Rolland, “Optimization of freeform lightpipes for light-emitting-diode projectors,” Appl. Opt. 47(7), 957–966 (2008). [CrossRef]
21. X. Mao, J. Li, F. Wang, R. Gao, X. Li, and Y. Xie, “Fast design method of smooth freeform lens with an arbitrary aperture for collimated beam shaping,” Appl. Opt. 58(10), 2512–2521 (2019). [CrossRef]
22. D. Ma, Z. Feng, and R. Liang, “Deconvolution method in designing freeform lens array for structured light illumination,” Appl. Opt. 54(5), 1114–1117 (2015). [CrossRef]
23. S. Wei, Z. Zhu, W. Li, and D. Ma, “Compact freeform illumination optics design by deblurring the response of extended sources,” Opt. Lett. 46(11), 2770–2773 (2021). [CrossRef]
24. P. Ge, K. Zhang, L. Mao, Y. Zhang, and H. Xu, “Design of a freeform lens for structured light illumination by deconvolution algorithm and feedback modification,” Light. Res. Technol. 51(1), 154–163 (2019). [CrossRef]
25. M. Brand and D. A. Birch, “Freeform irradiance tailoring for light fields,” Opt. Express 27(12), A611–A619 (2019). [CrossRef]
26. D. A. Birch and M. Brand, “Design of freeforms to uniformly illuminate polygonal targets from extended sources via edge ray mapping,” Appl. Opt. 59(22), 6490–6496 (2020). [CrossRef]
27. R. Winston, J. C. Mi nano, and P. G. Benitez, Nonimaging optics (Elsevier, 2005).
28. Z. Ding, L. Yang, R. Wu, Z. Zheng, M. Wang, P. Zeng, F. Wu, Y. Li, and R. Liang, “Designing compact and ultra-efficient illumination lenses with prescribed irradiance properties for extended light sources,” Opt. Commun. 465, 125601 (2020). [CrossRef]
